C₂×C₄₇×D₆₉ #27188449

order	12972
isoclinism	isoclinic class of D₆₉
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₄₇×D₆₉
unambiguous semidirect product	(C₂×C₃)⋊(D₂₃×C₄₇)
unambiguous semidirect product	(C₂×C₂₃)⋊(D₃×C₄₇)

Splittings

Fitting, socle	C₂×C₃×C₂₃×C₄₇ ⋊ C₂
derived, Hall	C₃×C₂₃ ⋊ C₂²×C₄₇
Hall	C₃×C₂₃×C₄₇ ⋊ C₂²
Hall	C₂₃×C₄₇ ⋊ C₂×D₃
Hall	C₃×C₄₇ ⋊ C₂×D₂₃
Sylow	C₂₃ ⋊ C₂×D₃×C₄₇
Sylow	C₃ ⋊ C₂×D₂₃×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄₇	D₆₉
commutator, cosocle, hypocenter	C₃×C₂₃	C₂²×C₄₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₂₃×C₄₇	C₂
LCS	G ▷ C₃×C₂₃
UCS	C₁ ◁ C₂×C₄₇

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6486

A minimal faithful permutation representation — 2 generators on 75 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,25,26)(74,75)

Conjugacy classes & character table

3384 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12972, 29);

C2×C47×D69 #27188449